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Is there any way to automatically make this double-triangulated plane as desired

from this:

default-triangulated plane

I guess it should be called "double-triangulation" or "cross-triangulation", but there's no trace of it in the docs nor anywhere on the Internet.

I need to split in two all the longest individual edges of triangulated mesh, and connect resulting vertices to closest vertices already exist, so the triangles I've recieved during the first triangulation were splitted in two. Thanks in advance.

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In blender it's called "Poke faces", but you have to perform it on quads to get the shape you are mentioning.

Here's a tut: https://www.youtube.com/watch?v=2Z2x3Wvzono

So:

  1. run the "Tris to Quads" operator on all the faces
  2. then run the "Poke Faces"

enter image description here

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Dissolve and Poke.

enter image description here

Select a long edge, the select similar by length to select all.

XLimited dissolve.

Finally poke the faces.

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